| L(s) = 1 | + 3·3-s + 5-s + 7-s + 6·9-s − 11-s − 6·13-s + 3·15-s + 3·17-s − 5·19-s + 3·21-s − 2·23-s + 25-s + 9·27-s − 5·29-s + 5·31-s − 3·33-s + 35-s − 37-s − 18·39-s − 2·41-s + 12·43-s + 6·45-s − 2·47-s − 6·49-s + 9·51-s − 13·53-s − 55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s + 0.377·7-s + 2·9-s − 0.301·11-s − 1.66·13-s + 0.774·15-s + 0.727·17-s − 1.14·19-s + 0.654·21-s − 0.417·23-s + 1/5·25-s + 1.73·27-s − 0.928·29-s + 0.898·31-s − 0.522·33-s + 0.169·35-s − 0.164·37-s − 2.88·39-s − 0.312·41-s + 1.82·43-s + 0.894·45-s − 0.291·47-s − 6/7·49-s + 1.26·51-s − 1.78·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.491010532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.491010532\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.89766511338015233329150568707, −9.786029738762734281399683357155, −9.483006760033315076271652715999, −8.241837378038505346249898701453, −7.79929638968846771610897877135, −6.73950043879569860448501959504, −5.18236780553163224027614577017, −4.07821246743313595622649858247, −2.77079934605572561635474291231, −1.97350758509779452319901265349,
1.97350758509779452319901265349, 2.77079934605572561635474291231, 4.07821246743313595622649858247, 5.18236780553163224027614577017, 6.73950043879569860448501959504, 7.79929638968846771610897877135, 8.241837378038505346249898701453, 9.483006760033315076271652715999, 9.786029738762734281399683357155, 10.89766511338015233329150568707
